Technical Report MSC-2011-19

Title: Topics in over-parametrization variational methods
Authors: Shem-Tov Shachar
Supervisors: Alfred M. Bruckstein
Abstract: This thesis discusses a variational methodology which involves local modeling of data from noisy samples, combined with global model parameter regularization. This methodology enables the reconstruction of local parameterized model describing the signal while addressing it’s main goal. We show that this methodology encompasses many previously proposed algorithms, from the celebrated moving least squares methods to the globally optimal over-parametrization methods. However, the unified look at the range of problems and methods previously considered also suggests a wealth of novel global functionals and local modeling possibilities. Over-parametrization is a general variational methodology, which enables prior knowledge about the properties of the problem domain to be readily incorporated by means of a set of ”basis” or ”dictionary functions”. The novelty of this methodology is in the fact that the regularization term is designed to penalize for deviation from the model which describes the domain, whereas common functionals penalize for deviation in the signal itself. This methodology yields excellent results for noise removing and for optical flow estimation. Specifically, we discuss two application of the proposed variational methodology. We begin by showing that the over-parameterized functional regularization may be expanded into an implicit segmentation process. Using a motion model which imposes a stereoscopic constraint, we show that this functional generates state of the art optical flow results. We then propose to incorporate a new non-local variational functional into the methodology, and show that it greatly improves robustness and accuracy in local model recovery compared to previous methods. The proposed methodology may be viewed as a basis for a general framework for addressing a variety of common research domains in signal and image processing and analysis, such as denoising, adaptive smoothing, reconstruction and segmentation.
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